Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,19,15,21,30,20)(11,17,23,35,24,37,39,36)(12,25,27,34,28,33,22,16)(13,29,26,18,31,38,40,32);(1,2,3,7,4,6,8,5)(9,11,13,22,30,39,40,28,15,24,31,27,14,23,26,12)(10,16,32,37,19,25,29,36,21,34,18,17,20,33,38,35);(9,12,26,23,14,27,31,24,15,28,40,39,30,22,13,11)(10,17,29,16,20,36,32,33,21,37,38,34,19,35,18,25)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48404);